Section: New Results

Lung and respiration modeling

Participants : Grégory Arbia, Laurent Boudin, Muriel Boulakia, Benoit Fabrèges, Miguel Ángel Fernández Varela, Jean-Frédéric Gerbeau, Céline Grandmont, Stéphane Liwarek, Jessica Oakes, Ayman Moussa, Irène Vignon-Clementel.

• In [65] we are interested in the mathematical modeling of the deformation of the human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma is a foam-like elastic material containing millions of air-filled alveoli connected by a tree- shaped network of airways. In this study, the parenchyma is governed by the linearized elasticity equations and the air movement in the tree by the Poiseuille law in each airway. The geometric arrangement of the alveoli is assumed to be periodic with a small period. We use the two-scale convergence theory to study the asymptotic behavior as the period goes to zero. The effect of the network of airways is described by a nonlocal operator and we propose a simple geometrical setting for which we show that this operator converges. We identify in the limit the equations modeling the homogenized behavior under an abstract convergence condition on this nonlocal operator. We derive some mechanical properties of the limit material by studying the homogenized equations: the limit model is nonlocal both in space and time if the parenchyma material is considered compressible, but only in space if it is incompressible. Finally, we propose a numerical method to solve the homogenized equations and we study numerically a few properties of the homogenized parenchyma model.

• In [30] we present a calculation of the functioning of an acinus at exercise. We show that, given the geometry and the breathing dynamics of real acini, respiration can be correlated to a single equivalent parameter that we call the integrative permeability. We find that both $V_{O2max}$ and PAO2 depend on this permeability in a non-linear manner.

• In [19] , In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim and under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed.

• In [18] ,In this work, we investigate the asymptotic behaviour of the solutions to the non-reactive fully elastic Boltzmann equations for mixtures in the diffusive scaling. We deal with cross sections such as hard spheres or cut-off power law potentials. We use Hilbert expansions near the common thermodynamic equilibrium granted by the H-theorem. The lower-order non trivial equality obtained from the Boltzmann equations leads to a linear functional equation in the velocity variable which is solved thanks to the Fredholm alternative. Since we consider multicomponent mixtures, the classical techniques introduced by Grad cannot be applied, and we propose a new method to treat the terms involving particles with different masses. The next-order equality in the Hilbert expansion then allows to write the macroscopic continuity equations for each component of the mixture.

• In [38] , In this paper we introduce a PDE system which aims at describing the dynamics of a dispersed phase of particles moving into an incompressible perfect fluid, in two space dimensions. The system couples a Vlasov-type equation and an Euler-type equation: the fluid acts on the dispersed phase through a gyroscopic force whereas the latter contributes to the vorticity of the former. First we give a Dobrushin type derivation of the system as a mean-field limit of a PDE system which describes the dynamics of a finite number of massive pointwise particles moving into an incompressible perfect fluid. This last system is itself inferred from a joint work of the second author with O. Glass and C. Lacave, where the system for one massive pointwise particle was derived as the limit of the motion of a solid body when the body shrinks to a point with fixed mass and circulation. Then we deal with the well-posedness issues including the existence of weak solutions. Next we exhibit the Hamiltonian structure of the system and finally, we study the behavior of the system in the limit where the mass of the particles vanishes.

• In [39] we solved for the airflow and aerosol particle distribution in healthy and emphysematous rat lungs. Following our preliminary work in [78] , we first estimated the respiratory resistance and compliance parameters from pressure measurements taken during ventilation experiments performed in healthy and emphysematous rats. Next, the 3D Navier-Stokes equations were solved in a Magnetic Resonance derived airway geometry coupled to 0D models at the boundaries leading to the five rat lobes. The multiscale 3D-0D simulations enabled consistent pressure and airflow results, unlike what was found when a constant pressure was described at the boundaries. Aerosolized particles were tracked throughout inspiration and the effects of particle size and gravity were studied. Healthy, homogeneous and heterogeneous disease cases were assessed. Once available, these in-silico predictions may be compared to experimental deposition data.